on \(H^\mathbb {Z}\times K\) . Then \(p\)  is a semiconjugacy from \((H^\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z},T^\mathrm {sg},\beta ^\mathrm {sg})\)  to \((H^\mathbb {Z}\times K,\tilde{T}^\mathrm {sg},\tilde{\beta }^\mathrm {sg})\) .
The fiber measure \((\delta _{k_{w^-}}+\delta _{-k_{w^-}})/2 \)  is actually the measure \((\nu _K)_w\)  for the unique \(\mu \) -stationary measure \(\nu _K\)  on \(K\)  and \((\nu _K)_w\)  is the limit of \(b_{-1}\cdots b_{-n}\nu _K\)  for \(\mu ^Z\) -a.e.  \(w\) . (This measure \((\nu _K)_w\)  is a lift of the Dirac mass \(\delta _{\xi (w^-)}\)  on \(H/Q_H\) . Since \(\nu _K\)  is unique, we can verify that the limiting measure has equal mass on two preimages). By [1]}, since \(\nu _K\)  is \(\mu \) -ergodic, we know that \(\tilde{\beta }^\mathrm {sg}\)  is \(\tilde{T}^\mathrm {sg}\)  ergodic. Then from the semiconjugacy \(p\) , we obtain that \(\beta ^\mathrm {sg}\)  is \(T^\mathrm {sg}\)  ergodic. The proof is complete.
\(\Box \)
